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C5×C29×C22⋊ΓL(1,𝔽8) #27131812

order12180
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C29 · Schur multiplier (H2) = C2

Aliases

direct productC5×C29×C22⋊ΓL(1,𝔽8)
unambiguous semidirect productC22⋊(C5×ΓL(1,𝔽8)×C29)
unambiguous semidirect productC7⋊(C5×A4×C29)
unambiguous semidirect product(C22×C5)⋊(ΓL(1,𝔽8)×C29)
unambiguous semidirect product(C5×C7)⋊(A4×C29)
unambiguous semidirect product(C22×C29)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×C29)⋊(C5×A4)
unambiguous semidirect product(C22×C5×C29)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C5×C7×C29)⋊A4

Splittings

Fitting, socle, HallC22×C5×C7×C29 ⋊ C3
derived, HallC22×C7 ⋊ C3×C5×C29
HallC5×C7×C29 ⋊ A4
HallC22×C7×C29 ⋊ C3×C5
HallC22×C5×C29 ⋊ ΓL(1,𝔽8)
HallC7×C29 ⋊ C5×A4
HallC22×C5×C7 ⋊ C3×C29
HallC22×C29 ⋊ C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ A4×C29
HallC22×C5 ⋊ ΓL(1,𝔽8)×C29
SylowC7 ⋊ C5×A4×C29
SylowC22 ⋊ C5×ΓL(1,𝔽8)×C29

Subgroups

subgroupG/H
center, hypercenterC5×C29C22⋊ΓL(1,𝔽8)
commutator, hypocenterC22×C7C3×C5×C29
FrattiniC1G
Fitting, socleC22×C5×C7×C29C3
LCSG ▷ C22×C7
UCSC1 ◁ C5×C29

40 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1740 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×A4×ΓL(1,𝔽8)×C29, C3×C5×C29×C22⋊ΓL(1,𝔽8)
maximal normal subgroup ofC5×A4×ΓL(1,𝔽8)×C29, C3×C5×C29×C22⋊ΓL(1,𝔽8)
maximal quotient ofC5×C29×C22⋊(C7⋊C9), C3×C5×C29×C22⋊ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12180, 147);